A one-per-cent distortion, and a million galaxies to see it

A cluster of galaxies bends the light of everything behind it. Where the alignment is close enough the effect is spectacular — arcs, rings, multiple images of one source — and the geometry of those images is a mass measurement.

That regime covers a tiny fraction of the sky and a tiny fraction of any cluster’s extent. Almost everywhere the deflection is far too small to produce a second image, and what it does instead is stretch each background galaxy by about a per cent, tangentially around the mass.

A per cent is not a detectable distortion of anything whose undistorted shape is unknown. Galaxies are not round: their ellipticities scatter with a dispersion of about 0.3 per component, so the signal is thirty times smaller than the noise on any single object.

The two quantities #

A lens does two things to a small background image. It changes its size, and it changes its shape.

The size change is the convergence $\kappa$, an isotropic magnification. The shape change is the shear $\gamma$, a two-component quantity with a magnitude and an orientation — and because an ellipse maps to itself under a rotation of 180°, the orientation enters doubled, exactly as a polarisation does.

The convergence has a clean physical meaning:

$$\kappa = \frac{\Sigma}{\Sigma_{\text{cr}}}, \qquad \Sigma_{\text{cr}} = \frac{c^2}{4\pi G}\frac{D_s}{D_l D_{ls}},$$

the projected surface density of the lens in units of a critical surface density set entirely by the geometry. That combination of distances is why weak lensing needs redshifts: without knowing where the source is, the conversion from a measured shear to a mass is unavailable. What is actually measured is neither $\gamma$ nor $\kappa$ but the reduced shear $g = \gamma/(1-\kappa)$, because a magnified image is measured as it appears. The two differ by ten per cent where $\kappa = 0.1$, which matters at the level modern surveys work to.

The estimator, and its noise #

If a galaxy’s intrinsic ellipticity is $\varepsilon^{\text{int}}$ and the reduced shear is $g$, the observed ellipticity is very nearly $\varepsilon = \varepsilon^{\text{int}} + g$. Averaging over many galaxies, the intrinsic term averages to zero — provided galaxies are not intrinsically aligned, which is an assumption and a known systematic — and

$$\langle\varepsilon\rangle = g.$$

The error on that average is $\sigma_\varepsilon/\sqrt{N}$ with $\sigma_\varepsilon \approx 0.3$, and that single expression is the design document for every weak-lensing survey ever proposed.

The consequences are all arithmetic. A one-per-cent shear at signal-to-noise ten needs about ninety thousand galaxies in the bin. The number available is the survey’s depth in galaxies per square arcminute times the area of the annulus, so depth and radial resolution trade against each other directly, and the depth is bought with exposure time.

It is also why the systematic requirement is so severe. Below about $10^{-3}$ in shear, the instrument’s own point-spread function stops being negligible — and averaging more galaxies makes an instrumental distortion no smaller at all.

That is a different relationship between effort and precision from the one most of observational astronomy has. A photometric measurement improves as the square root of the exposure until it reaches a calibration floor, and the floor can usually be lowered by working harder at the calibration. Here the floor is set by how well an optical system’s distortion can be characterised across a wide field over years, and the honest answer is that nobody knows how low it can be pushed. Every generation of survey has found the previous generation’s systematic floor to be higher than claimed.

Measuring a shape at all #

Before any of the statistics, there is a measurement problem that occupies most of the effort in a real analysis: extracting an ellipticity from a faint, small, blurred image.

The galaxies used are typically a fraction of an arcsecond across and blurred by a point-spread function of comparable size, detected at a signal-to-noise of ten or twenty. Their apparent shape is therefore the true shape convolved with the PSF and then sampled by pixels and buried in noise, and recovering an unbiased estimate of the pre-convolution ellipticity from that is genuinely hard.

Two biases dominate and both are multiplicative in the shear.

Noise bias. Ellipticity is a nonlinear function of the pixel values, so noise does not average out of it: a symmetric error in the pixels produces an asymmetric error in the derived shape. The effect scales as the inverse square of the signal-to-noise, and at the depths surveys work to it is a per-cent-level bias on a per-cent-level signal.

Model bias. Fitting an elliptical profile of an assumed form to a galaxy that is not that shape biases the answer by an amount depending on how wrong the assumption is — and real galaxies have bulges, discs, spiral arms and companions.

Both are calibrated rather than derived: enormous suites of simulated images, with known input shears, are pushed through the same pipeline, and the recovered shear is compared with the input. The correction factor is typically a few per cent and is quoted with its own uncertainty. A modern shear catalogue is a measurement multiplied by a number obtained from simulations, and saying so is part of the result.

The profile, and the null test #

Around a spherical lens the shear is tangential, and its azimuthal average has an exact interpretation:

$$\langle\gamma_t\rangle(R) = \frac{\bar{\Sigma}(<R) - \Sigma(R)}{\Sigma_{\text{cr}}},$$

the difference between the mean surface density inside a radius and the density at it. That combination — the excess surface density — is what a weak-lensing measurement delivers, and it is not the same as the surface density itself.

That last point is the field’s standard null test and it deserves its own paragraph. Decompose each galaxy’s ellipticity into a tangential component and one at 45° to it — the cross component. Gravitational lensing by any mass distribution whatever produces no cross component in the azimuthal average; it is a parity argument rather than a statement about any particular lens.

So a non-zero cross signal is an instrument. Measuring it costs nothing, it uses the same data, and it is the reason a weak-lensing mass can be believed at all.

There is a second null of the same kind, and it is arguably stronger. Rotate the positions of the sources rather than their shapes — measure the tangential shear around random points in the field instead of around the cluster — and the answer must be zero, because there is nothing there. Any residual is a measure of how much of the signal is being produced by the survey’s own geometry: field edges, masked regions around bright stars, and the varying depth across a mosaic. A null test that uses the real data and a fake centre is the cheapest systematic check in the subject, and it catches a class of error that the cross component does not.

What it is for #

Three things, in rising order of ambition.

Weighing clusters. A cluster weighed three ways — by galaxy velocities, by X-ray gas, and by lensing — gives three numbers that need not agree, and the first two both assume the cluster is relaxed. That difference produced the single most-cited image in the dark-matter argument. In the Bullet Cluster two clusters have passed through each other; the X-ray gas, which is collisional, was stripped and left in the middle, while the galaxies passed through. The lensing mass follows the galaxies, not the gas — so the dominant mass component is collisionless and is not the gas, which is most of the baryons.

The strength of that argument is worth stating precisely, because it is often overstated. It does not measure how much dark matter there is; the lensing reconstruction is not accurate enough for that. What it does is separate the mass from the light-emitting baryons spatially, which no rotation curve can, because in a galaxy the two are concentric. A measurement that separates two things that are ordinarily in the same place is worth more than a more precise measurement of either, and that is the whole of what a colliding cluster provides.

Mapping. Because the shear can be measured everywhere rather than only where arcs happen to form, a lensing survey produces a map of projected mass over degrees of sky, including regions with no light in them at all. Cosmology. The same distortion is produced by the large-scale structure between here and any distant galaxy, at an amplitude of about a per cent — cosmic shear. Correlating the shapes of galaxies as a function of their angular separation measures the amplitude of the matter power spectrum along the line of sight, integrated with a geometric weight, and the result is a constraint on the combination usually written $S_8$.

The other half of the lensing signal #

Shear is one of the two things a lens does and it gets nearly all the attention. The other — the magnification — is measurable too, and the reason it is used far less is instructive.

A lens magnifies, so a background source appears brighter than it would otherwise be. That pushes faint sources over a survey’s detection limit and increases the counts. It also stretches the sky, which spreads the same sources over a larger apparent area and decreases the counts. The two effects act in opposite directions and their net depends on the slope of the source counts: where the count of sources brighter than a limit rises steeply with the limit, the brightening wins and there is an excess behind a lens; where it rises slowly, the dilution wins and there is a deficit.

That sign flip is the measurement’s most attractive feature, because it is a signature nothing else has. A systematic that produced a spurious excess of sources behind a mass would have to produce a deficit for a differently selected sample at the same position, which no plausible instrumental effect does.

Its weakness is noise. Number counts fluctuate as the square root of the count, and the fractional signal is a few per cent — so a magnification measurement on a given field is several times noisier than a shear measurement on the same field, and it needs a source population whose count slope is known well enough to predict the sign and size.

What makes it worth the trouble is that it breaks the degeneracy shear cannot. A uniform sheet of convergence changes no shape and does change every magnitude, so a measurement sensitive to the magnification is sensitive to exactly the component the shear is blind to. Combining the two removes the mass-sheet transformation and turns a relative mass profile into an absolute one.

The two observables are complementary in the strict sense: one measures the traceless part of the distortion and the other the trace, and neither can be recovered from the other. That the field has leaned so heavily on the first is a statement about which is easier to measure rather than about which carries more.

What it still cannot do #

The mass-sheet degeneracy. Adding a uniform sheet of surface density to a lens and rescaling the sources changes no observed shape at all. So a shear measurement determines the mass profile only up to that transformation, and breaking it requires the magnification — which is measurable, through the number counts of background sources, and far noisier than the shear.

Photometric redshifts. $\Sigma_{\text{cr}}$ depends on the distances to the lens and to each source, and a survey of a hundred million galaxies cannot obtain spectra. The redshifts come from broadband colours, with errors of a few per cent and catastrophic failures at the per-cent level, and the resulting uncertainty in $\Sigma_{\text{cr}}$ is now the leading systematic in most analyses.

Intrinsic alignments. The estimator assumes the intrinsic ellipticities average to zero. Galaxies that formed in the same tidal field do not: they are weakly aligned with each other and with the structure around them, which mimics a shear signal at the ten-per-cent level and has to be modelled rather than measured away. The alignment has a distinctive redshift dependence — it is a local effect, so it correlates pairs at the same distance while lensing correlates a background pair with a foreground mass — and separating the two is done by exploiting exactly that difference.

Weighing clusters to count them #

The most consequential use of weak lensing is not the measurement of any individual cluster but the calibration of a relation, and setting that out explains why so much effort goes into the systematics.

Counting clusters as a function of mass and redshift measures how fast structure grew, and the count is straightforward: a survey finds clusters by their galaxies, by their X-ray emission or by their shadow on the microwave background, and the number found in each redshift interval is a number. What is not straightforward is the mass, because none of those three detection methods measures one.

Each supplies an observable — a richness, an X-ray luminosity, a distortion amplitude — which correlates with mass with some scatter. Turning a count in observable into a count in mass requires that relation, and the relation has to be calibrated, and the calibration has to be done by something that measures mass without assuming equilibrium.

That is what lensing is for. Stacking the shear around many clusters binned by observable gives the mean mass in each bin, directly, and the mean mass as a function of observable is the calibration. The scatter about the relation matters as much as its slope, because a steeply falling mass function convolved with a scatter produces more clusters at high observable than the mean relation predicts — the same Eddington bias that inflates the bright end of any steeply falling count.

The chain from a lensing measurement to a cosmological parameter is therefore long: shapes to shear to stacked profiles to mean masses to a mass–observable relation to a corrected count to a growth rate. Every systematic in this essay propagates through all of it, which is why an analysis quoting a per-cent constraint on structure growth spends most of its length on shape measurement and photometric redshifts.

It is also why the discrepancy mentioned earlier is hard to resolve. A cluster count and a cosmic shear measurement both report slightly less structure than the microwave background predicts, and both depend on the same shape catalogue — so the two are not the independent confirmations they superficially appear to be.

The measurement is a signal-to-noise argument with three terms in it, and each of them is worth reading at a second value.

What a hundred thousand galaxies buys, and what it does not #

It is worth putting the survey arithmetic beside a concrete question, because the scaling has a consequence that is easy to state and easy to forget.

Suppose the aim is a mass profile for one cluster in eight radial bins, each measured to ten per cent. At a shear of one per cent in the outer bins, ten per cent precision needs a shear error of $10^{-3}$, which needs ninety thousand galaxies in that bin. Multiply by eight bins and the field has to supply of order half a million usable background galaxies.

At a survey depth of thirty galaxies per square arcminute — which is a good deep ground-based survey — that is about five thousand square arcminutes, or one and a half square degrees, which is far larger than a cluster. So the outer bins are limited not by the cluster but by how much sky the instrument can cover to that depth.

The way out is not more sky per cluster but more clusters. Stacking the shear profiles of a thousand clusters gives the mean profile of the population to the required precision with an entirely feasible survey, at the price of learning nothing about any individual object.

That trade runs through the whole subject. Weak lensing is a statistical instrument in two senses at once — it averages over galaxies to see one cluster, and it averages over clusters to see a profile — and the quantity it delivers well is always a population mean. The individual masses that appear in catalogues carry uncertainties of tens of per cent, and the cosmological results rest on the stacks.

And the field itself for a lens half as strong, which is the regime almost every real cluster occupies.

Where this ladder goes next #

This rung has established the estimator, the noise floor that designs the survey, and the null test that makes the result credible.

The rung above is the combination of shear with magnification and with galaxy clustering — the so-called three-by-two-point analysis — which breaks the mass-sheet degeneracy and constrains the relation between galaxies and mass at the same time.

Beside it lies cluster cosmology: counting clusters as a function of mass and redshift is a sensitive probe of structure growth, and the count is only as good as the masses, which is to say it is a lensing measurement wearing a different hat.

And below it, the habit: a signal far below the noise on any one object is still a measurement if the noise is independent and the signal is not. Everything in this essay follows from that one asymmetry, and from being willing to average a million things to find one per cent.